2. Capital Asset pricing Model

Dr Youchang WuDr. Youchang Wu

WS 2007

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Efficient frontier in the presence of a risk-free asset

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Capital market lineCapital market lineWhen a risk free asset exists i e when a capital marketWhen a risk-free asset exists, i.e., when a capital market is introduced, the efficient frontier is linear.This linear frontier is called capital market linepThe capital market line touches the efficient frontier of the risky assets only at the tangency portfolioThe optimal portfolio of any investor with mean varianceThe optimal portfolio of any investor with mean-variance preference can be constructed using the risk-free asset and the tangency portfolio, which contains only risky assets (Two fund separation)assets (Two-fund separation)All investors with a mean-variance preference, independent of their risk attitudes, hold the same pportfolio of risky assets. The risk attitude only affects the relative weights of the risk-free asset and the risky portfolio

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risk free asset and the risky portfolio

Market price of riskMarket price of riskThe equation for the capital market lineThe equation for the capital market line

)()(

)( pfT

fP rrrE

rrE σ−

+=

The slope of the capital market line represents the best risk-return t d ff th t i il bl th k t

)()(

)( pT

fP rσ

trade-off that is available on the marketWithout the risk-free asset, the marginal rate of substitution between risk and return will be different across investorsAft i t d i th i k f t it ill b th f llAfter introducing the risk-free asset, it will be the same for all investors.Almost all investors benefit from introducing the risk-free asset (capital market)(capital market)Reminiscent of the Fisher Separation Theorem?

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Remaining questionsRemaining questions

What would be the tangency portfolio in equilibrium? CML describes the risk-return relation for all efficient portfolios, but what is the equilibrium relation between risk and return for inefficient portfolios or individual assets?What if the risk-free asset does not exist?

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CAPMCAPMIf b d h ld h i k f li h h i k f liIf everybody holds the same risky portfolio, then the risky portfolio must be the MARKET portfolio, i.e., the value-weighted portfolio of all securities (demand must equal supply in equilibrium!).It follows that the market portfolio is a frontier portfolio.According to Property III of portfolio frontier (discussed last time), we must havemust have

)(/),(],)([)( 2mmjjfmjfj rrrCOVwhererrErrE σββ =−+=

This is the famous CAPM independently derived by Treynor(1961), Sharpe (1964), Lintner(1965) and Mossin (1966)E( ) i ll d k t i k i A t‘ i k i iE(rm)-rf is called market risk premium. An asset‘s risk premium is given by its beta time the market risk premium.

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Underlying assumptionsUnderlying assumptions

Mean-variance preferenceHomogeneous beliefs among investorsHomogeneous beliefs among investors regarding the planning horizon and the distribution of security returnsdistribution of security returnsNo market frictions: no transaction costs, no tax, no restriction on short selling, no information cost etc.information cost etc.

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Intuition for CAPMIntuition for CAPMSince everybody holds the market portfolio the risk of an individualSince everybody holds the market portfolio, the risk of an individual asset is characterized by its contribution to variance of the market portfolio instead of its own variance.The contribution of an individual security to the variance of marketThe contribution of an individual security to the variance of market portfolio is determined by its covariance with the market portfolio

)(2n n r∂σ

The part of an asset‘s risk that is correlated with the market portfolio

),(2)(),()(1 1

2mi

i

mjij

i jim rrCOV

wrrrCOVwwr =

∂∂

=>= ∑∑= =

σσ

The part of an asset s risk that is correlated with the market portfolio, the systematic risk, cannot be diversified away; thus, investors need to be compensated for bearing it.The part of an asset’s risk that is not correlated with the marketThe part of an asset s risk that is not correlated with the market portfolio, the unsystematic risk, can be diversified away; thus, bearing unsystematic risk need not be rewarded, and therefore, an asset’s unsystematic risk does not affect its risk premium.

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CAPM vs Property IIICAPM vs Property IIIProperty III is a mathematical property of portfolio frontier thatProperty III is a mathematical property of portfolio frontier that holds for any return distribution.CAPM is an asset pricing relation derived using economic

i It ifi h i k d t l t d ireasoning. It specifies how risk and return are related in equilibrium.Asset returns under CAPM are not exogenously given. They

d l d t i d i k t ilib iare endogenously determined in market equilibrium.– If the CAPM does not hold, i.e., market portfolio is not

mean-variance efficient, then demand does not equal l f tsupply for some assets.

– The prices of such assets as well as other assets must change, therefore the whole return distribution will change

– This process will continue until we converge to an return distribution that makes the market portfolio mean-variance efficient.

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Security market lineSecurity market line

E(r) E(r)Capital market line

Security market line

E(r) E(r)

E(rm)E(rm)

rfrf

σ ββ=1

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SML vs CMLSML vs CML

Capital market line describes the efficient frontier in the presence of a risk-free passet. It specifies the equilibrium relation between return and total risk of efficientbetween return and total risk of efficient portfolio.SSecurity market line describes the equilibrium relation between return and qsystematic risk for all assets or portfolios

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Portfolio betaPortfolio beta

CAPM h ld b th f i di id l t dCAPM holds both for individual assets and portfolios of assetsThe beta of a portfolio is simply the weighted average beta of each individual g gassets in the portfolio, this is because

nn

),(),(),(11

mi

iimi

iimp rrCOVwrrwCOVrrCOV ∑∑==

==

where wi is the weight of asset i in portfolio p.

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Zero beta CAPMZero-beta CAPMWhat if there is no risk free asset?What if there is no risk-free asset?Each investor will hold a different frontier portfolio depending on his own risk attitudep gAccording to Property I of portfolio frontier, the aggregate of each investor‘s portfolio, i.e., the market portfolio, is also a frontier portfolioalso a frontier portfolioBy Property III of portfolio frontier, it follows that the linear relation between expected return and beta with respect to the market portfolio must holdrespect to the market portfolio must hold.This argument leads to the zero-beta CAPM (Black 1972):)

where rz is the return of a zero-beta portfolio

)]()([)()( zmjzj rErErErE −+= β

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where rz is the return of a zero beta portfolio

Implications for investmentImplications for investmentCAPM l th t i t t i i i t tCAPM solves three most important issues in investment simultaneously:– Where to invest? Invest in the market portfolio and risk-freeWhere to invest? Invest in the market portfolio and risk free

asset!– How to value an asset? Estimated the beta of this asset and

discount all the future cash flows generated by this asset using adiscount all the future cash flows generated by this asset using a discount rate given by CAPM!

– How to evaluate investment performance? Adjust the performance by beta!performance by beta!

Not surprisingly this is regarded as one of the greatest results in financeBut how does it fit the real world? Numerous studies have tried to answer this question

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Earlier empirical tests of CAPMEarlier empirical tests of CAPM Cross sectional testCross-sectional test– First estimate the beta by running the following time-serise

regression rr εβα ++=– Then run the following (out of sample) cross-sectional regression

jtmtjjjt rr εβα ++=

eCHARr +++= γβγγwhere CHAR is a charateristic of stock j unrelated to CAPM such firm size, idiosyncratic risk

jjjj eCHARr +++= 210 γβγγ

– CAPM predicts that γ0=risk free rate, γ1 =market risk premium, and γ2 =0

– Tests are usually based on portfolio returns to avoid t i ti t d b tmeasurement errors in estimated betas.

– Supportive results are reported by Fama and MacBeth (1974)

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Earlier empirical tests of CAPM (2)Earlier empirical tests of CAPM (2)

Time series test

rrrr εβα +−+=− )(

Prediction of CAPM: α is zero for every stock

jtfmtjjfjt rrrr εβα +−+=− )(

Prediction of CAPM: αj is zero for every stock or portfolioBlack Jensen and Scholes (1972) reject theBlack, Jensen and Scholes (1972) reject the standard CAPM in favor of the zero-beta CAPMCAPM

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Roll´s critiqueRoll s critique Roll (1977) argues that CAPM is inherently untestableRoll (1977) argues that CAPM is inherently untestable– The only economic prediction of CAPM is that the

market portfolio is mean-variance efficient p– The linear return/beta relation can be found in any

sample irrespective of how returns are determined in the market (All we need is to identify an indexthe market (All we need is to identify an index portfolio which is ex post efficient)

– All existing tests only tell us whether the market proxy used by researchers are efficient or not they sayused by researchers are efficient or not, they say nothing about the efficiency of the market portfolio itself

– A true market portflio should include all assets (human capital etc) and is unobservable, therefore CAPM is untestable

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Roll´s critique (2)Roll s critique (2)

Roll (1978) further argues that using CAPM to measure performance is problematic– If performance is measure relative to an index that is

ex post efficient, then from PIII of porftolio frontier, all tf li ill b th it k t li (portfolios will be on the security market line (no

performance)If performance is measured relative to an ex post– If performance is measured relative to an ex post inefficient index, then any ranking of portfolio performance is possible depending on which p p p ginefficient index has been chosen

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Findings from more recent testsFindings from more recent testsTh l ti b t b t d t i k tThe relation between beta and return is weak at best (Fama and French 1992)Si ff t ll t k t f l t kSize effect: small stocks outperform large stocks (Banz 1981,Fama and French 1992)B k t k t ff t t k ith hi h b k tBook-to-market effect: stocks with high book-to-market equity ratios outperform stocks with low book to market ratios (Fama and French 1992)book-to-market ratios (Fama and French 1992)Momentum effect: Stocks outperforming in the last 3 12 months tend to outperform in thelast 3-12 months tend to outperform in the following 3-12 months (Jegadeesh and Titman 1993)

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1993).

Possible explanations for empirical shortcomings of CAPMM tMeasurement errors– Measurement errors in beta– Measurement errors in expected return (survivorship bias)Measurement errors in expected return (survivorship bias)

Behavioral biases of investors– Small investors are subject to many behavioral biases– Institutional investors are subject to agency issues

Missing risk factorsPeople do not just hold bonds and stocks– People do not just hold bonds and stocks

– Therefore they do not just care about an asset‘s covariance with a market index

– They also care about its covariance with other components (such as human capital) of their „true“ portfolio

– This means we have to consider other risk factors as well

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SummarySummary

CAPM is an elegant model that gives simple answers to several key issues in p yfinanceThe real prediction of CAPM is that theThe real prediction of CAPM is that the market portfolio is mean-variance efficient.Empirical results should be interpreted with great caution in the light of Roll´swith great caution in the light of Roll s critique

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